first chunk of compute commands to be converted to use embedded math
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\documentclass[12pt]{article}
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|
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\begin{document}
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|
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$$
|
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CS = \sum_{i = 1}^{N/2} | \vec{R}_i + \vec{R}_{i+N/2} |^2
|
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$$
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\end{document}
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\documentclass[12pt,article]{article}
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\usepackage{indentfirst}
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\usepackage{amsmath}
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\begin{document}
|
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|
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\begin{eqnarray*}
|
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r_{c}^{fcc} & = & \frac{1}{2} \left(\frac{\sqrt{2}}{2} + 1\right) \mathrm{a} \simeq 0.8536 \:\mathrm{a} \\
|
||||
r_{c}^{bcc} & = & \frac{1}{2}(\sqrt{2} + 1) \mathrm{a} \simeq 1.207 \:\mathrm{a} \\
|
||||
r_{c}^{hcp} & = & \frac{1}{2}\left(1+\sqrt{\frac{4+2x^{2}}{3}}\right) \mathrm{a}
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\end{eqnarray*}
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\end{document}
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\documentclass[12pt,article]{article}
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\usepackage{indentfirst}
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\usepackage{amsmath}
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\begin{document}
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$$
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Rc + Rs > 2*{\rm cutoff}
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$$
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\end{document}
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\documentclass[12pt,article]{article}
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\usepackage{indentfirst}
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\usepackage{amsmath}
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\begin{document}
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$$
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Rc + Rs > 2*{\rm cutoff}
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$$
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\end{document}
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\documentclass[12pt]{article}
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\begin{document}
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$$
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Q_{i} = \frac{1}{n_i}\sum_{j = 1}^{n_i} | \sum_{k = 1}^{n_{ij}} \vec{R}_{ik} + \vec{R}_{jk} |^2
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$$
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\end{document}
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\documentstyle[12pt]{article}
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\begin{document}
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$$
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{R_g}^2 = \frac{1}{M} \sum_i m_i (r_i - r_{cm})^2
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$$
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\end{document}
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\documentclass[12pt]{article}
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\pagestyle{empty}
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\begin{document}
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|
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\begin{eqnarray*}
|
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c = l_z - 0.5(l_y+l_x) \\
|
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b = l_y - l_x \\
|
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k = \frac{3}{2} \frac{l_x^2+l_y^2+l_z^2}{(l_x+l_y+l_z)^2} - \frac{1}{2}
|
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\end{eqnarray*}
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\end{document}
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|
@ -52,20 +52,23 @@ in the specified compute group.
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This parameter is computed using the following formula from
|
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:ref:`(Kelchner) <Kelchner>`
|
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|
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.. image:: Eqs/centro_symmetry.jpg
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:align: center
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.. math::
|
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|
||||
where the *N* nearest neighbors of each atom are identified and Ri and
|
||||
Ri+N/2 are vectors from the central atom to a particular pair of
|
||||
nearest neighbors. There are N\*(N-1)/2 possible neighbor pairs that
|
||||
can contribute to this formula. The quantity in the sum is computed
|
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for each, and the N/2 smallest are used. This will typically be for
|
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pairs of atoms in symmetrically opposite positions with respect to the
|
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central atom; hence the i+N/2 notation.
|
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CS = \sum_{i = 1}^{N/2} | \vec{R}_i + \vec{R}_{i+N/2} |^2
|
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|
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*N* is an input parameter, which should be set to correspond to the
|
||||
number of nearest neighbors in the underlying lattice of atoms. If
|
||||
the keyword *fcc* or *bcc* is used, *N* is set to 12 and 8
|
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|
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where the :math:`N` nearest neighbors of each atom are identified and
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:math:`\vec{R}_i` and :math:`\vec{R}_{i+N/2}` are vectors from the
|
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central atom to a particular pair of nearest neighbors. There are
|
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:math:`N (N-1)/2` possible neighbor pairs that can contribute to this
|
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formula. The quantity in the sum is computed for each, and the
|
||||
:math:`N/2` smallest are used. This will typically be for pairs of
|
||||
atoms in symmetrically opposite positions with respect to the central
|
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atom; hence the :math:`i+N/2` notation.
|
||||
|
||||
:math:`N` is an input parameter, which should be set to correspond to
|
||||
the number of nearest neighbors in the underlying lattice of atoms.
|
||||
If the keyword *fcc* or *bcc* is used, *N* is set to 12 and 8
|
||||
respectively. More generally, *N* can be set to a positive, even
|
||||
integer.
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|
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|
@ -74,9 +77,9 @@ lattice, the centro-symmetry parameter will be 0. It will be near 0
|
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for small thermal perturbations of a perfect lattice. If a point
|
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defect exists, the symmetry is broken, and the parameter will be a
|
||||
larger positive value. An atom at a surface will have a large
|
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positive parameter. If the atom does not have *N* neighbors (within
|
||||
the potential cutoff), then its centro-symmetry parameter is set to
|
||||
0.0.
|
||||
positive parameter. If the atom does not have :math:`N` neighbors
|
||||
(within the potential cutoff), then its centro-symmetry parameter is
|
||||
set to 0.0.
|
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|
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If the keyword *axes* has the setting *yes*\ , then this compute also
|
||||
estimates three symmetry axes for each atom's local neighborhood. The
|
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|
@ -95,7 +98,7 @@ of any atom.
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|
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Only atoms within the cutoff of the pairwise neighbor list are
|
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considered as possible neighbors. Atoms not in the compute group are
|
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included in the *N* neighbors used in this calculation.
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included in the :math:`N` neighbors used in this calculation.
|
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|
||||
The neighbor list needed to compute this quantity is constructed each
|
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time the calculation is performed (e.g. each time a snapshot of atoms
|
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|
|
|
@ -51,8 +51,12 @@ E.g. 12 nearest neighbor for perfect FCC and HCP crystals, 14 nearest
|
|||
neighbors for perfect BCC crystals. These formulas can be used to
|
||||
obtain a good cutoff distance:
|
||||
|
||||
.. image:: Eqs/cna_cutoff1.jpg
|
||||
:align: center
|
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.. math::
|
||||
|
||||
r_{c}^{fcc} = & \frac{1}{2} \left(\frac{\sqrt{2}}{2} + 1\right) \mathrm{a} \simeq 0.8536 \:\mathrm{a} \\
|
||||
r_{c}^{bcc} = & \frac{1}{2}(\sqrt{2} + 1) \mathrm{a} \simeq 1.207 \:\mathrm{a} \\
|
||||
r_{c}^{hcp} = & \frac{1}{2}\left(1+\sqrt{\frac{4+2x^{2}}{3}}\right) \mathrm{a}
|
||||
|
||||
|
||||
where a is the lattice constant for the crystal structure concerned
|
||||
and in the HCP case, x = (c/a) / 1.633, where 1.633 is the ideal c/a
|
||||
|
@ -62,10 +66,13 @@ Also note that since the CNA calculation in LAMMPS uses the neighbors
|
|||
of an owned atom to find the nearest neighbors of a ghost atom, the
|
||||
following relation should also be satisfied:
|
||||
|
||||
.. image:: Eqs/cna_cutoff2.jpg
|
||||
:align: center
|
||||
.. math::
|
||||
|
||||
where Rc is the cutoff distance of the potential, Rs is the skin
|
||||
r_c + r_s > 2*{\rm cutoff}
|
||||
|
||||
|
||||
where :math:`r_c` is the cutoff distance of the potential, :math:`r_s`
|
||||
is the skin
|
||||
distance as specified by the :doc:`neighbor <neighbor>` command, and
|
||||
cutoff is the argument used with the compute cna/atom command. LAMMPS
|
||||
will issue a warning if this is not the case.
|
||||
|
|
|
@ -40,13 +40,16 @@ only be performed on single component systems.
|
|||
This parameter is computed using the following formula from
|
||||
:ref:`(Tsuzuki) <Tsuzuki2>`
|
||||
|
||||
.. image:: Eqs/cnp_eq.jpg
|
||||
:align: center
|
||||
.. math::
|
||||
|
||||
where the index *j* goes over the *n*\ i nearest neighbors of atom
|
||||
*i*\ , and the index *k* goes over the *n*\ ij common nearest neighbors
|
||||
between atom *i* and atom *j*\ . Rik and Rjk are the vectors connecting atom
|
||||
*k* to atoms *i* and *j*\ . The quantity in the double sum is computed
|
||||
Q_{i} = \frac{1}{n_i}\sum_{j = 1}^{n_i} | \sum_{k = 1}^{n_{ij}} \vec{R}_{ik} + \vec{R}_{jk} |^2
|
||||
|
||||
|
||||
where the index *j* goes over the :math:`n_i` nearest neighbors of atom
|
||||
*i*\ , and the index *k* goes over the :math:`n_{ij}` common nearest neighbors
|
||||
between atom *i* and atom *j*\ . :math:`\vec{R}_{ik}` and
|
||||
:math:`\vec{R}_{jk}` are the vectors connecting atom *k* to atoms *i*
|
||||
and *j*\ . The quantity in the double sum is computed
|
||||
for each atom.
|
||||
|
||||
The CNP calculation is sensitive to the specified cutoff value.
|
||||
|
@ -56,8 +59,12 @@ E.g. 12 nearest neighbor for perfect FCC and HCP crystals, 14 nearest
|
|||
neighbors for perfect BCC crystals. These formulas can be used to
|
||||
obtain a good cutoff distance:
|
||||
|
||||
.. image:: Eqs/cnp_cutoff.jpg
|
||||
:align: center
|
||||
.. math::
|
||||
|
||||
r_{c}^{fcc} = & \frac{1}{2} \left(\frac{\sqrt{2}}{2} + 1\right) \mathrm{a} \simeq 0.8536 \:\mathrm{a} \\
|
||||
r_{c}^{bcc} = & \frac{1}{2}(\sqrt{2} + 1) \mathrm{a} \simeq 1.207 \:\mathrm{a} \\
|
||||
r_{c}^{hcp} = & \frac{1}{2}\left(1+\sqrt{\frac{4+2x^{2}}{3}}\right) \mathrm{a}
|
||||
|
||||
|
||||
where a is the lattice constant for the crystal structure concerned
|
||||
and in the HCP case, x = (c/a) / 1.633, where 1.633 is the ideal c/a
|
||||
|
@ -67,10 +74,13 @@ Also note that since the CNP calculation in LAMMPS uses the neighbors
|
|||
of an owned atom to find the nearest neighbors of a ghost atom, the
|
||||
following relation should also be satisfied:
|
||||
|
||||
.. image:: Eqs/cnp_cutoff2.jpg
|
||||
:align: center
|
||||
.. math::
|
||||
|
||||
where Rc is the cutoff distance of the potential, Rs is the skin
|
||||
r_c + r_s > 2*{\rm cutoff}
|
||||
|
||||
|
||||
where :math:`r_c` is the cutoff distance of the potential, :math:`r_s` is
|
||||
the skin
|
||||
distance as specified by the :doc:`neighbor <neighbor>` command, and
|
||||
cutoff is the argument used with the compute cnp/atom command. LAMMPS
|
||||
will issue a warning if this is not the case.
|
||||
|
|
|
@ -26,19 +26,26 @@ Description
|
|||
"""""""""""
|
||||
|
||||
Define a computation that accumulates the total internal conductive
|
||||
energy (U\_cond), the total internal mechanical energy (U\_mech), the
|
||||
total chemical energy (U\_chem) and the *harmonic* average of the internal
|
||||
temperature (dpdTheta) for the entire system of particles. See the
|
||||
energy (:math:`U^{cond}`), the total internal mechanical energy
|
||||
(:math:`U^{mech}`), the total chemical energy (:math:`U^{chem}`)
|
||||
and the *harmonic* average of the internal temperature (:math:`\theta_{avg}`)
|
||||
for the entire system of particles. See the
|
||||
:doc:`compute dpd/atom <compute_dpd_atom>` command if you want
|
||||
per-particle internal energies and internal temperatures.
|
||||
|
||||
The system internal properties are computed according to the following
|
||||
relations:
|
||||
|
||||
.. image:: Eqs/compute_dpd.jpg
|
||||
:align: center
|
||||
.. math::
|
||||
|
||||
where N is the number of particles in the system
|
||||
U^{cond} = & \displaystyle\sum_{i=1}^{N} u_{i}^{cond} \\
|
||||
U^{mech} = & \displaystyle\sum_{i=1}^{N} u_{i}^{mech} \\
|
||||
U^{chem} = & \displaystyle\sum_{i=1}^{N} u_{i}^{chem} \\
|
||||
U = & \displaystyle\sum_{i=1}^{N} (u_{i}^{cond} + u_{i}^{mech} + u_{i}^{chem}) \\
|
||||
\theta_{avg} = & (\frac{1}{N}\displaystyle\sum_{i=1}^{N} \frac{1}{\theta_{i}})^{-1} \\
|
||||
|
||||
|
||||
where :math:`N` is the number of particles in the system
|
||||
|
||||
|
||||
----------
|
||||
|
@ -46,8 +53,9 @@ where N is the number of particles in the system
|
|||
|
||||
**Output info:**
|
||||
|
||||
This compute calculates a global vector of length 5 (U\_cond, U\_mech,
|
||||
U\_chem, dpdTheta, N\_particles), which can be accessed by indices 1-5.
|
||||
This compute calculates a global vector of length 5 (:math:`U^{cond}`,
|
||||
:math:`U^{mech}`, :math:`U^{chem}`, :math:`\theta_{avg}`, :math:`N`),
|
||||
which can be accessed by indices 1-5.
|
||||
See the :doc:`Howto output <Howto_output>` doc page for an overview of
|
||||
LAMMPS output options.
|
||||
|
||||
|
|
|
@ -23,10 +23,10 @@ Description
|
|||
"""""""""""
|
||||
|
||||
Define a computation that accesses the per-particle internal
|
||||
conductive energy (u\_cond), internal mechanical energy (u\_mech),
|
||||
internal chemical energy (u\_chem) and
|
||||
internal temperatures (dpdTheta) for each particle in a group. See
|
||||
the :doc:`compute dpd <compute_dpd>` command if you want the total
|
||||
conductive energy (:math:`u^{cond}`), internal mechanical
|
||||
energy (:math:`u^{mech}`), internal chemical energy (:math:`u^{chem}`)
|
||||
and internal temperatures (:math:`\theta`) for each particle in a group.
|
||||
See the :doc:`compute dpd <compute_dpd>` command if you want the total
|
||||
internal conductive energy, the total internal mechanical energy, the
|
||||
total chemical energy and
|
||||
average internal temperature of the entire system or group of dpd
|
||||
|
@ -34,14 +34,16 @@ particles.
|
|||
|
||||
**Output info:**
|
||||
|
||||
This compute calculates a per-particle array with 4 columns (u\_cond,
|
||||
u\_mech, u\_chem, dpdTheta), which can be accessed by indices 1-4 by any
|
||||
This compute calculates a per-particle array with 4 columns (:math:`u^{cond}`,
|
||||
:math:`u^{mech}`, :math:`u^{chem}`, :math:`\theta`), which can be accessed
|
||||
by indices 1-4 by any
|
||||
command that uses per-particle values from a compute as input. See
|
||||
the :doc:`Howto output <Howto_output>` doc page for an overview of
|
||||
LAMMPS output options.
|
||||
|
||||
The per-particle array values will be in energy (u\_cond, u\_mech, u\_chem)
|
||||
and temperature (dpdTheta) :doc:`units <units>`.
|
||||
The per-particle array values will be in energy (:math:`u^{cond}`,
|
||||
:math:`u^{mech}`, :math:`u^{chem}`)
|
||||
and temperature (:math:`theta`) :doc:`units <units>`.
|
||||
|
||||
Restrictions
|
||||
""""""""""""
|
||||
|
|
|
@ -53,27 +53,33 @@ information about the solid structure is required.
|
|||
This parameter for atom i is computed using the following formula from
|
||||
:ref:`(Piaggi) <Piaggi>` and :ref:`(Nettleton) <Nettleton>` ,
|
||||
|
||||
.. image:: Eqs/pair_entropy.jpg
|
||||
:align: center
|
||||
.. math::
|
||||
|
||||
s_S^i=-2\pi\rho k_B \int\limits_0^{r_m} \left [ g(r) \ln g(r) - g(r) + 1 \right ] r^2 dr
|
||||
|
||||
|
||||
where r is a distance, g(r) is the radial distribution function of atom
|
||||
i and rho is the density of the system. The g(r) computed for each
|
||||
atom i can be noisy and therefore it is smoothed using:
|
||||
|
||||
.. image:: Eqs/pair_entropy2.jpg
|
||||
:align: center
|
||||
.. math::
|
||||
|
||||
where the sum in j goes through the neighbors of atom i, and sigma is a
|
||||
parameter to control the smoothing.
|
||||
g_m^i(r) = \frac{1}{4 \pi \rho r^2} \sum\limits_{j} \frac{1}{\sqrt{2 \pi \sigma^2}} e^{-(r-r_{ij})^2/(2\sigma^2)}
|
||||
|
||||
The input parameters are *sigma* the smoothing parameter, and the
|
||||
*cutoff* for the calculation of g(r).
|
||||
|
||||
where the sum in j goes through the neighbors of atom i, and :math:`\sigma`
|
||||
is a parameter to control the smoothing.
|
||||
|
||||
The input parameters are *sigma* the smoothing parameter :math:`\sigma`,
|
||||
and the *cutoff* for the calculation of g(r).
|
||||
|
||||
If the keyword *avg* has the setting *yes*\ , then this compute also
|
||||
averages the parameter over the neighbors of atom i according to:
|
||||
|
||||
.. image:: Eqs/pair_entropy3.jpg
|
||||
:align: center
|
||||
.. math::
|
||||
|
||||
\left< s_S^i \right> = \frac{\sum_j s_S^j + s_S^i}{N + 1}
|
||||
|
||||
|
||||
where the sum j goes over the neighbors of atom i and N is the number
|
||||
of neighbors. This procedure provides a sharper distinction between
|
||||
|
|
|
@ -70,14 +70,18 @@ initial interactions of the atoms that will undergo perturbation, and
|
|||
a term :math:`U_1` corresponding to the final interactions of
|
||||
these atoms:
|
||||
|
||||
.. image:: Eqs/compute_fep_u.jpg
|
||||
:align: center
|
||||
.. math::
|
||||
|
||||
U(\lambda) = U_{\mathrm{bg}} + U_1(\lambda) + U_0(\lambda)
|
||||
|
||||
A coupling parameter :math:`\lambda` varying from 0 to 1 connects the
|
||||
reference and perturbed systems:
|
||||
|
||||
.. image:: Eqs/compute_fep_lambda.jpg
|
||||
:align: center
|
||||
.. math::
|
||||
|
||||
\lambda = 0 \quad\Rightarrow\quad U = U_{\mathrm{bg}} + U_0 \\
|
||||
\lambda = 1 \quad\Rightarrow\quad U = U_{\mathrm{bg}} + U_1
|
||||
|
||||
|
||||
It is possible but not necessary that the coupling parameter (or a
|
||||
function thereof) appears as a multiplication factor of the potential
|
||||
|
@ -89,16 +93,23 @@ This command can be combined with :doc:`fix adapt <fix_adapt>` to
|
|||
perform multistage free-energy perturbation calculations along
|
||||
stepwise alchemical transformations during a simulation run:
|
||||
|
||||
.. image:: Eqs/compute_fep_fep.jpg
|
||||
:align: center
|
||||
.. math::
|
||||
|
||||
\Delta_0^1 A = \sum_{i=0}^{n-1} \Delta_{\lambda_i}^{\lambda_{i+1}} A =
|
||||
- kT \sum_{i=0}^{n-1} \ln \left< \exp \left( - \frac{U(\lambda_{i+1}) -
|
||||
U(\lambda_i)}{kT} \right) \right>_{\lambda_i}
|
||||
|
||||
This compute is suitable for the finite-difference thermodynamic
|
||||
integration (FDTI) method :ref:`(Mezei) <Mezei>`, which is based on an
|
||||
evaluation of the numerical derivative of the free energy by a
|
||||
perturbation method using a very small :math:`\delta`:
|
||||
|
||||
.. image:: Eqs/compute_fep_fdti.jpg
|
||||
:align: center
|
||||
.. math::
|
||||
|
||||
\Delta_0^1 A = \int_{\lambda=0}^{\lambda=1} \left( \frac{\partial
|
||||
A(\lambda)}{\partial\lambda} \right)_\lambda \mathrm{d}\lambda
|
||||
\approx \sum_{i=0}^{n-1} w_i \frac{A(\lambda_{i} + \delta) -
|
||||
A(\lambda_i)}{\delta}
|
||||
|
||||
where :math:`w_i` are weights of a numerical quadrature. The :doc:`fix adapt <fix_adapt>` command can be used to define the stages of
|
||||
:math:`\lambda` at which the derivative is calculated and averaged.
|
||||
|
@ -109,16 +120,23 @@ choosing a very small perturbation :math:`\delta` the thermodynamic
|
|||
integration method can be implemented using a numerical evaluation of
|
||||
the derivative of the potential energy with respect to :math:`\lambda`:
|
||||
|
||||
.. image:: Eqs/compute_fep_ti.jpg
|
||||
:align: center
|
||||
.. math::
|
||||
|
||||
\Delta_0^1 A = \int_{\lambda=0}^{\lambda=1} \left< \frac{\partial
|
||||
U(\lambda)}{\partial\lambda} \right>_\lambda \mathrm{d}\lambda
|
||||
\approx \sum_{i=0}^{n-1} w_i \left< \frac{U(\lambda_{i} + \delta) -
|
||||
U(\lambda_i)}{\delta} \right>_{\lambda_i}
|
||||
|
||||
|
||||
|
||||
Another technique to calculate free energy differences is the
|
||||
acceptance ratio method :ref:`(Bennet) <Bennet>`, which can be implemented
|
||||
by calculating the potential energy differences with :math:`\delta` = 1.0 on
|
||||
both the forward and reverse routes:
|
||||
|
||||
.. image:: Eqs/compute_fep_bar.jpg
|
||||
:align: center
|
||||
.. math::
|
||||
|
||||
\left< \frac{1}{1 + \exp\left[\left(U_1 - U_0 - \Delta_0^1A \right) /kT \right]} \right>_0 = \left< \frac{1}{1 + \exp\left[\left(U_0 - U_1 + \Delta_0^1A \right) /kT \right]} \right>_1
|
||||
|
||||
The value of the free energy difference is determined by numerical
|
||||
root finding to establish the equality.
|
||||
|
@ -265,9 +283,11 @@ If the keyword *volume* = *yes*\ , then the Boltzmann term is multiplied
|
|||
by the volume so that correct ensemble averaging can be performed over
|
||||
trajectories during which the volume fluctuates or changes :ref:`(Allen and Tildesley) <AllenTildesley>`:
|
||||
|
||||
.. image:: Eqs/compute_fep_vol.jpg
|
||||
:align: center
|
||||
.. math::
|
||||
|
||||
\Delta_0^1 A = - kT \sum_{i=0}^{n-1} \ln \frac{\left< V \exp \left( -
|
||||
\frac{U(\lambda_{i+1}) - U(\lambda_i)}{kT} \right)
|
||||
\right>_{\lambda_i}}{\left< V \right>_{\lambda_i}}
|
||||
|
||||
----------
|
||||
|
||||
|
|
|
@ -32,24 +32,27 @@ periodic boundaries.
|
|||
Rg is a measure of the size of the group of atoms, and is computed as
|
||||
the square root of the Rg\^2 value in this formula
|
||||
|
||||
.. image:: Eqs/compute_gyration.jpg
|
||||
:align: center
|
||||
.. math::
|
||||
|
||||
where M is the total mass of the group, Rcm is the center-of-mass
|
||||
position of the group, and the sum is over all atoms in the group.
|
||||
{R_g}^2 = \frac{1}{M} \sum_i m_i (r_i - r_{cm})^2
|
||||
|
||||
A Rg\^2 tensor, stored as a 6-element vector, is also calculated by
|
||||
this compute. The formula for the components of the tensor is the
|
||||
same as the above formula, except that (Ri - Rcm)\^2 is replaced by
|
||||
(Rix - Rcmx) \* (Riy - Rcmy) for the xy component, etc. The 6
|
||||
components of the vector are ordered xx, yy, zz, xy, xz, yz. Note
|
||||
that unlike the scalar Rg, each of the 6 values of the tensor is
|
||||
effectively a "squared" value, since the cross-terms may be negative
|
||||
|
||||
where :math:`M` is the total mass of the group, :math:`r_{cm}` is the
|
||||
center-of-mass position of the group, and the sum is over all atoms in
|
||||
the group.
|
||||
|
||||
A :math:`{R_g}^2` tensor, stored as a 6-element vector, is also calculated
|
||||
by this compute. The formula for the components of the tensor is the
|
||||
same as the above formula, except that :math:`(r_i - r_{cm})^2` is replaced
|
||||
by :math:`(r_{i,x} - r_{cm,x}) \cdot (r_{i,y} - r_{cm,y})` for the xy component,
|
||||
and so on. The 6 components of the vector are ordered xx, yy, zz, xy, xz, yz.
|
||||
Note that unlike the scalar :math:`R_g`, each of the 6 values of the tensor
|
||||
is effectively a "squared" value, since the cross-terms may be negative
|
||||
and taking a sqrt() would be invalid.
|
||||
|
||||
.. note::
|
||||
|
||||
The coordinates of an atom contribute to Rg in "unwrapped" form,
|
||||
The coordinates of an atom contribute to :math:`R_g` in "unwrapped" form,
|
||||
by using the image flags associated with each atom. See the :doc:`dump custom <dump>` command for a discussion of "unwrapped" coordinates.
|
||||
See the Atoms section of the :doc:`read_data <read_data>` command for a
|
||||
discussion of image flags and how they are set for each atom. You can
|
||||
|
@ -58,8 +61,8 @@ and taking a sqrt() would be invalid.
|
|||
|
||||
**Output info:**
|
||||
|
||||
This compute calculates a global scalar (Rg) and a global vector of
|
||||
length 6 (Rg\^2 tensor), which can be accessed by indices 1-6. These
|
||||
This compute calculates a global scalar (:math:`R_g`) and a global vector of
|
||||
length 6 (:math:`{R_g}^2` tensor), which can be accessed by indices 1-6. These
|
||||
values can be used by any command that uses a global scalar value or
|
||||
vector values from a compute as input. See the :doc:`Howto output <Howto_output>` doc page for an overview of LAMMPS output
|
||||
options.
|
||||
|
|
|
@ -52,11 +52,13 @@ boundaries.
|
|||
Rg is a measure of the size of a chunk, and is computed by this
|
||||
formula
|
||||
|
||||
.. image:: Eqs/compute_gyration.jpg
|
||||
:align: center
|
||||
.. math::
|
||||
|
||||
where M is the total mass of the chunk, Rcm is the center-of-mass
|
||||
position of the chunk, and the sum is over all atoms in the
|
||||
{R_g}^2 = \frac{1}{M} \sum_i m_i (r_i - r_{cm})^2
|
||||
|
||||
|
||||
where :math:`M` is the total mass of the chunk, :math:`r_{cm}` is
|
||||
the center-of-mass position of the chunk, and the sum is over all atoms in the
|
||||
chunk.
|
||||
|
||||
Note that only atoms in the specified group contribute to the
|
||||
|
@ -70,14 +72,16 @@ non-zero chunk IDs.
|
|||
If the *tensor* keyword is specified, then the scalar Rg value is not
|
||||
calculated, but an Rg tensor is instead calculated for each chunk.
|
||||
The formula for the components of the tensor is the same as the above
|
||||
formula, except that (Ri - Rcm)\^2 is replaced by (Rix - Rcmx) \* (Riy -
|
||||
Rcmy) for the xy component, etc. The 6 components of the tensor are
|
||||
formula, except that :math:`(r_i - r_{cm})^2` is replaced by
|
||||
:math:`(r_{i,x} - r_{cm,x}) \cdot (r_{i,y} - r_{cm,y})` for the xy
|
||||
component, and so on. The 6 components of the tensor are
|
||||
ordered xx, yy, zz, xy, xz, yz.
|
||||
|
||||
.. note::
|
||||
|
||||
The coordinates of an atom contribute to Rg in "unwrapped" form,
|
||||
by using the image flags associated with each atom. See the :doc:`dump custom <dump>` command for a discussion of "unwrapped" coordinates.
|
||||
The coordinates of an atom contribute to :math:`R_g` in "unwrapped" form,
|
||||
by using the image flags associated with each atom. See the :doc:`dump custom <dump>`
|
||||
command for a discussion of "unwrapped" coordinates.
|
||||
See the Atoms section of the :doc:`read_data <read_data>` command for a
|
||||
discussion of image flags and how they are set for each atom. You can
|
||||
reset the image flags (e.g. to 0) before invoking this compute by
|
||||
|
|
|
@ -33,10 +33,14 @@ due to atoms passing through periodic boundaries.
|
|||
The three computed shape parameters are the asphericity, b, the acylindricity, c,
|
||||
and the relative shape anisotropy, k:
|
||||
|
||||
.. image:: Eqs/compute_shape_parameters.jpg
|
||||
:align: center
|
||||
.. math::
|
||||
|
||||
where lx <= ly <= lz are the three eigenvalues of the gyration tensor. A general description
|
||||
c = & l_z - 0.5(l_y+l_x) \\
|
||||
b = & l_y - l_x \\
|
||||
k = & \frac{3}{2} \frac{l_x^2+l_y^2+l_z^2}{(l_x+l_y+l_z)^2} - \frac{1}{2}
|
||||
|
||||
|
||||
where :math:`l_x` <= :math:`l_y` <= :math:`l_z` are the three eigenvalues of the gyration tensor. A general description
|
||||
of these parameters is provided in :ref:`(Mattice) <Mattice1>` while an application to polymer systems
|
||||
can be found in :ref:`(Theodorou) <Theodorou1>`.
|
||||
The asphericity is always non-negative and zero only when the three principal
|
||||
|
|
|
@ -33,10 +33,14 @@ all effects due to atoms passing through periodic boundaries.
|
|||
The three computed shape parameters are the asphericity, b, the acylindricity, c,
|
||||
and the relative shape anisotropy, k:
|
||||
|
||||
.. image:: Eqs/compute_shape_parameters.jpg
|
||||
:align: center
|
||||
.. math::
|
||||
|
||||
where lx <= ly <= lz are the three eigenvalues of the gyration tensor. A general description
|
||||
c = & l_z - 0.5(l_y+l_x) \\
|
||||
b = & l_y - l_x \\
|
||||
k = & \frac{3}{2} \frac{l_x^2+l_y^2+l_z^2}{(l_x+l_y+l_z)^2} - \frac{1}{2}
|
||||
|
||||
|
||||
where :math:`l_x` <= :math:`l_y` <= :math`l_z` are the three eigenvalues of the gyration tensor. A general description
|
||||
of these parameters is provided in :ref:`(Mattice) <Mattice2>` while an application to polymer systems
|
||||
can be found in :ref:`(Theodorou) <Theodorou2>`. The asphericity is always non-negative and zero
|
||||
only when the three principal moments are equal. This zero condition is met when the distribution
|
||||
|
|